J Stat Phys (2011) 145:734–744DOI 10.1007/s10955-011-0308-6

The Random Quadratic Assignment Problem

Gerald Paul · Jia Shao · H. Eugene Stanley

Received: 20 April 2011 / Accepted: 5 August 2011 / Published online: 2 September 2011© Springer Science+Business Media, LLC 2011

Abstract The quadratic assignment problem, QAP, is one of the most difficult of all com-binatorial optimization problems. Here, we use an abbreviated application of the statisticalmechanics replica method to study the asymptotic behavior of instances in which the en-tries of at least one of the two matrices that specify the problem are chosen from a randomdistribution P . Surprisingly, the QAP has not been studied before using the replica methoddespite the fact that the QAP was first proposed over 50 years ago and the replica methodwas developed over 30 years ago. We find simple forms for Cmin and Cmax, the costs of theminimal and maximum solutions respectively. Notable features of our results are the sym-metry of the results for Cmin and Cmax and their dependence on P only through its mean andstandard deviation, independent of the details of P .

Keywords Quadratic assignment problem · Replica method

1 Introduction

Optimal assignment of classes to classrooms [11], design of DNA microarrays [9], crossspecies gene analysis [19], creation of hospital layouts [13], and assignment of componentsto locations on circuit boards [26] are a few of the many problems which have been formu-lated as a quadratic assignment problem (QAP). The QAP is a combinatorial optimizationproblem first introduced by Koopmans and Beckmann [20]. It is NP-hard and is consideredto be one of the most difficult problems to be solved optimally. The problem was defined inthe following context: A set of N facilities are to be located at N locations. The quantity ofmaterials which flow between facilities i and j is Aij and the distance between locations i

and j is Bij . The problem is to assign to each location a single facility so as to minimize

G. Paul (!) · J. Shao · H.E. StanleyCenter for Polymer Studies and Dept. of Physics, Boston University, Boston, MA 02215, USAe-mail: gerryp@bu.edu

The Random Quadratic Assignment Problem 735

(or maximize) the cost

C =N!

i=1

N!

j=1

AijBp(i)p(j), (1)

where p(i) represents the location to which i is assigned.In addition to being important in its own right, the QAP includes such other combinatorial

optimization problems as the traveling salesman problem and graph partitioning as specialcases. There is an extensive literature which addresses the QAP and is reviewed in [3, 6,10, 17, 22, 24]. The QAP is exceedingly hard. With the exception of specially constructedcases, optimal algorithms have solved only relatively small instances with N ! 36. Variousheuristic approaches have been developed and applied to problems typically of size N " 100or less. By contrast, a traveling salesman problem consisting of almost 25,000 towns inSweden has been solved exactly [4].

Our approach makes use of the replica method of statistical mechanics. The replicamethod is notorious for the tremendously complicated calculations usually involved. How-ever, we take advantage of the fact that if only the form of the solution is desired, the calcula-tion is relatively straightforward. While we formulate the problem using the replica method,we proceed only through the first step which consists of averaging over the disorder repre-sented by the random matrix. At this point, using elementary arguments, we can infer theform of the minimum and maximum costs.

We find that in the asymptotic limit in which the size of the problem N # $, the costsof the minimum and maximum solutions, Cmin and Cmax respectively, are

Cmin = µAµBN2 % !Af (B)N3/2 (2)

Cmax = µAµBN2 + !Af (B)N3/2, (3)

where A is a matrix the elements of which are chosen from the random distribution P (Aij )

and the elements of B are arbitrary. Here µA and !A are the mean and standard deviationsof the distribution P (A); µB is the mean of the entries of B , and f is a function of B and N .Equations (2) and (3) hold under the condition that both matrices are dense (have O(N2)

non-zero entries). Our goal below is to argue for the form of (2) and (3). We do not attemptto determine the value of the functions f (B).

2 Relation to Previous Work

Previous work on asymptotic properties of random QAP instances has been generally limitedto the case in which the elements of both matrices are drawn from random (usually uniform)distributions [1, 2, 5, 7, 8, 14, 21, 25]. Here we consider the properties of solutions to theQAP under the requirement that the elements of only one of the matrices need be drawn fromany random distribution P . Burkard and Fincke [8] implicitly studied the case in which onlyone matrix is random. They proved rigorously that, under certain conditions, asymptoticallyas N # $, the ratio between Cmin and Cmax approaches 1, a result with which our (2) and(3) are consistent. Our work extends [8] by finding the closed form (2) and (3). Rhee [25]finds a closed form solution similar to (2) when both matrices are random and does not findthe explicit dependence on !A. Finally, statistical mechanics methods have been previouslyused to study the random QAP [2, 5] although to the best of our knowledge the replicamethod had not been used heretofore.

736 G. Paul et al.

3 Random Solution Cost

It is useful to first consider the solution for which p = p& is a random permutation. Becausethe elements of A are assigned randomly and since p&(i) and p&(j) are random, each Bij inthe sum is multiplied by a random value of Aij the average of which is µA. Hence, the costof a random permutation is

Crand = µA

N!

i,j=1

Bp&(i)p&(j) = µAµBN2. (4)

4 Replica Analysis

We now use the replica method to derive the form for Cmin and then derive the relationshipof Cmax to Cmin. Employing a Hamiltonian, H, defined as the QAP cost function our goal isto compute the partition function

Z =!

{p}exp

"% H

kT

#=

!

{p}exp

$

% 1kT

N!

i,j=1

AijBp(i)p(j)

%

(5)

and the free energy

% F

kT= lim

N#$lnZ (6)

where k and T are the Boltzmann constant and temperature respectively. Then,

Cmin = F(T = 0). (7)

Since the Hamiltonian includes a random matrix, A, we want to calculate the value of thefree energy F averaged over the disorder specified by the probability distribution P (A).However, averaging the log of the partition function is difficult. The replica method of sta-tistical mechanics [12] was introduced to make calculation of this average possible. Thereplica method has been used not just on models of physical systems (such as spin glasses[12, 18, 23]) but also on such combinatorial optimization problems as graph partitioningand the traveling salesman problem [15, 16, 23]. The calculation of the average of thepartition function is simplified using a mathematical identity known as the replica trick,ln(x) = limn#0(x

n % 1)/n. Then (6) becomes

% F

kT= lim

N#$limn#0

1n

&Zn % 1

', (8)

where

Zn =(!

{p1}exp

"%H({p1})

kT

#). . .

(!

{pn}exp

"%H({pn})

kT

#)(9)

and Zn '*

P (A)Zn dA denotes Zn averaged over the disorder. Here each Hamiltonianrepresents a replica of the original system and the sum over {p"} now denotes the sum overall permutations in all replicas.

The Random Quadratic Assignment Problem 737

In order to achieve physically sensible results with f ' F/N intensive, we require themean and standard deviation of P (A) to scale as (see [15, 18, 23])

µA = µA/N

!A = !A/(

N (10)

with µA and !A independent of N . In Appendix A we then find that

Zn = exp"%nµAµBN2

kT

#

)!

{p" }exp

$! 2

A

2(kT )2

!

i,j

+n!

"

Bp"(i)p"(j)

,2%

. (11)

As explained in Appendix B, (11) holds under the condition that neither A nor B is sparse.We can make the following observations based on (11) and (8):

– Consistent with (2), the dependence of F on A is only through µA and !A.– If !A = 0 and/or T # $, substituting (11) in (8) yields F = µAµBN2 which is the cost of

the random solution, (4). This is reasonable because (i) physically for high temperatureswe expect randomness and (ii) if !A = 0, all entries in A are identical and all permutationsyield the same costs.

We infer the form of F(T = 0) as follows:

– In (11) !A appears in the combination !A/T . Thus, from (8) we see that in the T # 0limit, only a term linear in !A can survive in F .

This linear dependence on !A as well as on µA is consistent with the simple casein which all of the elements in A are scaled by a constant, z, in which case !A # z!A

and µA # zµA. Clearly the optimal permutation is unchanged but the cost is also scaledby z. Thus, in this simple case, for any permutation (including the optimal one) the lineardependence on µA and !A must hold.

– Given that we obtained (11) by expanding in 1/(

N , we expect the second term in theexpressions for Cmin to be proportional N3/2 since the leading term is proportional to N2.

Given the considerations, above the only possible expression for the second term in F is!Af (B)N3/2 where f is a function of B only.

The form of Cmax follows directly as follows. Let Cmin(A,B) and Cmax(A,B) denote theoptimal minimum and maximum costs respectively of the QAP problem with matrices A

and B and let %A denote a matrix with elements %Aij . Since Cmax(A,B) = %Cmin(%A,B)

and since µ%A = %µA and !%A = !A, the form for Cmax in equation (3) follows directlyfrom the form for Cmin.

If the entries of B are also drawn from a random distribution as in previous work, it isstraightforward to show that

Cmin = µAµBN2 % c!A!BN3/2

Cmax = µAµBN2 + c!A!BN3/2(12)

where c is a constant independent of A and B and !B is the standard deviation of the entriesin B . The dependence of the N3/2 term on A and B only through their standard deviationshas not been shown before.

738 G. Paul et al.

Fig. 1 For an N = 100 QAPinstance consisting of an Amatrix with elements from aGaussian distribution and a Bmatrix representing atwo-dimensional grid, (from topto bottom), Cmax, Crand, andCmin versus standarddeviation !A . For a given !A ,Cmax and Cmin values areequidistant from Crand value

5 Numerical Results

To confirm our findings, we use the tabu search (TS) [27] heuristic to obtain approximatenumerical solutions for a number of QAP instances. We employ matrices of the types de-scribed in detail in Appendix D. We use the notation “A matrix type”-“B matrix type” tospecify a QAP instance.

In Fig. 1, we plot Cmin and Cmax versus !A for an instance of type Gaussian-Grid. Asexpected, the plots are linear in !A and the absolute values of Cmin and Cmax are equal for agiven ! . This is consistent with (2) and (3).

A stronger test is achieved by studying instances specified by a matrix that represents arandom graph of average degree k. In this case,

!A(k) =-

k(N % 1 % k) =.(

N % 12

)2

%(

k % N % 12

)2

(13)

which represents a circle with origin at ((N % 1)/2,0). In Fig. 2(a) we plot Cmin, Crand, andCmax versus k, 0 ! k ! N %1, for an instance of type Random-Grid. In order to illustrate thebehavior of Cmin and Cmax in more detail, in Fig. 2(b), we plot #Cmin/max ' Cmin/max %Crand.The solid line is an ellipse of the form

Ctheorymax/min = ±!A(k)f (B)N3/2 (14)

where f (B) is chosen to best fit of (14) to the data. The fit is consistent with the theory,exhibiting both the expected linear dependence of the optimal costs on !A(k) and the sym-metry represented by (2) and (3). In Fig. 3 we show similar plots for other varied QAPinstances.

We now study the dependence of #C on N . We treat instances in which the A matrix israndom or random regular and consider different types of B matrix. To compare results forinstances of different sizes, we define the normalized quantities #Cnorm and knorm

#Cnorm ' #C

µBN2; knorm ' k

N % 1. (15)

With this normalization we expect

#Cnorm * !AN%1/2. (16)

In Fig. 4(a), (c), and (e) we plot #Cnorm for various values of N for various instance types.We confirm the N%1/2 dependence by plotting

The Random Quadratic Assignment Problem 739

Fig. 2 For an N = 100 QAPinstance consisting of an Amatrix representing a randomgraph and B matrix representinga two-dimensional grid, (fromtop to bottom), (a) Cmax, Crand,and Cmin versus average degreek and (b) #Cmax and #Cminversus k. In this and all followingfigures, the upper and lowersemi-circles are the #Cmax and#Cmin plots, respectively. Thesolid circular line represents thetheoretical prediction

Fig. 3 For various N = 100 QAP instance, #Cmax and #Cmin versus k. The solid circular line representsthe theoretical prediction

#Ccollapsed ' #CnormN1/2 (17)

in Fig. 4(b), (d) and (f). The collapse is consistent with (16). Additional plots for otherinstance types are shown in Fig. 2 in the Supplement.

740 G. Paul et al.

Fig. 4 (a), (c), (e) Normalized #C versus normalized k for instance sizes N = 100 (light gray); N = 225for (a) and 200 for (c) and (e) (medium gray); and N = 400 (black). The right hand column contains thecorresponding collapsed plots

6 Discussion and Summary

With an innovative use of the statistical mechanics replica method involving only elementaryarguments and minimal mathematics, we determine the form of the values of the minimaland maximal costs for the random QAP. This is a significant advancement on a problem forwhich progress has been exceedingly slow. We find remarkably simple forms for Cmin andCmax providing insight into what have heretofore been isolated numerical results. This worknot only (i) presents concrete results for the QAP but also (ii) illustrates the power of thereplica method to unify and extend results on a very difficult problem.

Acknowledgements We thank S.V. Buldyrev for helpful discussions and the Defense Threat ReductionAgency (DTRA) for support.

The Random Quadratic Assignment Problem 741

Appendix A: Integration over Disorder

In the following we retain only terms which do not vanish in the N # $ limit. This isequivalent to retaining terms only to second order in Aij . Because we want to maintain theexponential form, we write

Zn =/

P (A)!

{p" }exp

$

% 1kT

n!

"=1

N!

i,j=1

AijBp"(i)p"(j)

%

dA

=!

{p"}exp

"ln

/P (A)e% 1

kT

0i,j

0n" Aij Bp" (i)p" (j) dA

#

=!

{p"}exp

"ln

1

i,j

/P (Aij )e

% 1kT

0n" Aij Bp" (i)p" (j) dAij

#

=!

{p"}exp

"!

i,j

ln/

P (Aij )e% 1

kT

0n" Aij Bp" (i)p" (j) dAij

#

=!

{p"}exp

"!

i,j

ln(1 + yij )

#(18)

where yij '*

P (Aij ) exp[% 1kT

0n" AijBp"(i)p"(j)]dAij % 1. Expanding yij to second order

in Aij we have:

yij */

P (Aij )

$

1 % 1kT

n!

"

AijBp"(i)p"(j) + (0n

" AijBp"(i)p"(j))2

2(kT )2

%

dAij % 1

= %µA

kT

n!

"

Bp"(i)p"(j) + µ2A

2(kT )2

+n!

"

Bp"(i)p"(j)

,2

(19)

where µA2 is the second moment (around zero) of P . Expanding ln(1 + yij ) to the secondorder in yij and substituting (19), we have

ln(1 + yij ) * %µA

kT

n!

"

Bp"(i)p"(j) + µ2A

2(kT )2

+n!

"

Bp"(i)p"(j)

,2

% 12

+µA

kT

n!

"

Bp"(i)p"(j)

,2

= %µA

kT

n!

"

Bp"(i)p"(j) + ! 2A

2(kT )2

+n!

"

Bp"(i)p"(j)

,2

(20)

where we have only retained terms to O(A2ij ). Finally we have

Zn =!

{p"}exp

$!

i,j

$

%µA

kT

n!

"

Bp"(i)p"(j) + ! 2A

2(kT )2

+n!

"

Bp"(i)p"(j)

,2%%

=!

{p"}exp

$

%µAµBnN2

kT+ ! 2

A

2(kT )2

!

i,j

+n!

"

Bp"(i)p"(j)

,2%

(21)

where we use the fact that0

i,j Bp"(i)p"(j) is independent of permutation.

742 G. Paul et al.

Appendix B: Requirements on A and B Matrices

That µA and !A scale as in (10) in order to achieve physically sensible results with f =F/N intensive (i.e. independent of N ) is required so that the quantities in the exponentsof (2) do not scale as N2 which would ultimately result in f scaling as N as N # $.The exponents will scale as N2 if A and B are dense (have O(N2) non-zero elements). If,however, either matrix is sparse the exponents will scale as N and the requirements on uA

and !A do not hold. We cannot then expand equation (18) in Aij as done in Appendix A.Thus our results do not apply to QAP instances in which both matrices are not dense, such asthe traveling salesman problem. Our results do apply to graph partitioning of dense graphsincluding random graphs in which the average degree k of the graph scales such that k/N isa constant—the case we have treated here (see also Refs. [15, 28]).

The requirement that A and B must be dense appears to be equivalent to a requirement,specified by Burkhard and Fincke [8] in which they studied asymptotic properties of prob-lems with sum (e.g. QAP) and bottleneck objective functions. They proved rigorously that,under certain conditions, asymptotically as N # $, the ratio between Cmin and Cmax ap-proaches 1, a result with which our (2) and (3) are consistent. A key condition is that |S|,the number of non-zero elements which contribute to the sum in the objective function,increases to infinity faster than the log of the number of feasible solutions |T |:

$0|S| % log |T | # $ as N # $ (22)

for a certain fixed constant $0 > 0.For the QAP, |T | * N ! and |S| * N2 if A and B are dense but |S| * N if either is sparse.

Equation (22) holds in the former case but not in the latter.Using statistical mechanical methods in Ref. [2], Albrecher et al. also make use of the

requirement of (22) to prove the results of Ref. [8] generalizing and repairing an earlierstatistical mechanical approach [5].

Appendix C: Relationship to Graph Partitioning

The problem of partitioning a graph into two subgraphs of size rN and (1 % r)N with theminimum number of edges between the two subgraphs can be represented as a QAP asfollows: One matrix, A, is the adjacency matrix of the graph to be partitioned. The othermatrix, B , the graph partitioning matrix, is the adjacency matrix for a bipartite graph inwhich edges are present between two sets of vertices; one set contains rN vertices and thesecond set contains (1 % r)N vertices. The QAP cost function is the cost of partitioning thegraph represented by A.

Appendix D: Matrix Types

We employ matrices of the following types:

– Uniform—the matrix elements are chosen from a uniform distribution on the interval[0,100].

– Gaussian—matrix elements are chosen from a Gaussian distribution with zero mean andstandard deviation ! .

The Random Quadratic Assignment Problem 743

– Half-Gaussian—matrix elements are chosen from a Gaussian distribution as above butonly elements with value greater or equal to zero are used.

– Random (graph)—the matrix is the adjacency matrix of a random graph with edgespresent with probability p. The average degree of the graph is k = pN .

– Random Regular (graph)—the matrix is the adjacency matrix of a random regular graphfor which all vertices are degree k.

– Grid—the matrix elements are the Euclidean distances between points in a two-dimensional square grid. The distances between adjacent points along the x and y axesare 100.

– Graph Partitioning—the matrix is the graph partitioning matrix described in Appendix C.

All matrices are symmetrical with zero diagonal. For the Random and Random Regularmatrices that represent graphs, we study cases of the graph degree k ranging from 0 toN % 1.

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